The orders mean the Louisiana-based Fourth Brigade of the 10th Mountain Division will deploy only 1300 soldiers to Afghanistan instead of all 4000 as previously scheduled, a senior military officer told The New York Times on Tuesday.
The troops staying at home will be on standby, the official told the newspaper, speaking on condition of anonymity.
Pentagon spokesman Larry Di Rita told the newspaper the decision to reduce a portion of the army unit scheduled to replace the 173rd Airborne Brigade, now in southern Afghanistan, was based on recommendations from the senior US commanders in Afghanistan and the Middle East.
Di Rita was quoted as saying: "The overall level of security forces in the country, Nato's role, and the political developments are all moving in the right direction."
A Pentagon spokesman was not immediately available for comment.
Rumsfeld signed the troop reduction orders on Monday and a formal Pentagon announcement is expected on Tuesday, The New York Times said.
The American troop reduction has been anticipated since Nato agreed to assume control of an American command in southern Afghanistan next year, the report said.
Nato is looking to raise its 9000-strong ISAF peacekeeping force to about 15,000 troops from early next year. It will spread its bases in the north and west, and the capital Kabul, to the more volatile south, a base for many insurgents.